Problem $1$: Suppose that a manager needs to order a component to satisfy next week$$s demand. The probability that there will be x $($integer$)$ units of demand is $1/20$ where $10<$ x $<=30.$

The manager can pay $$1$ per unit to order some components from a regular channel in advance. If it turns out that the amount is not sufficient to satisfy all demands, the manager needs to place an expediting orders at $$1\mathrm{.}8$ per unit to cover the shortage. Unused component can be sold in the second$-$hand market at $$0\mathrm{.}3$ per unit. Determine

$1.$The optimal number of units that the manager should order from the regular channel;